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Ukrainian Mathematical Journal

, Volume 68, Issue 12, pp 1849–1859 | Cite as

On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences

  • P. Das
  • E. Savas
Article

We consider the notion of generalized density, namely, the natural density of weight g recently introduced in [M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Acta Math. Hung., 147, No. 1, 97–115 (2015)] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results are also obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of M. Kücükaslan, U. Deger, and O. Dovgoshey, [Ukr. Math. J., 66, No. 5, 712–720 (2014)].

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References

  1. 1.
    M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, “Generalized kinds of density and the associated ideals,” Acta Math. Hungar., 147, No. 1, 97–115 (2015).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    G. S. Baranenkov, B. P. Demidovich, V. A. Efimenko, etc., Problems in Mathematical Analysis, Mir, Moscow (1976).Google Scholar
  3. 3.
    S. Bhunia, P. Das, and S. K. Pal, “Restricting statistical convergence,” Acta Math. Hungar., 134, No. 1-2, 153–161 (2012).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    V. Bilet, “Geodesic spaces tangent to metric spaces,” Ukr. Math. Zh., 64, No. 9, 1448–1456 (2012); English translation: Ukr. Math. J., 64, No. 9, 1273–1281 (2013).Google Scholar
  5. 5.
    V. Bilet and O. Dovgoshey, “Isometric embeddings of pretangent spaces in E n,” Bull. Belg. Math. Soc. Simon Stevin, 20, 91–110 (2013).MathSciNetMATHGoogle Scholar
  6. 6.
    R. Colak, “Statistical convergence of order α,” in: Modern Methods in Analysis and Its Applications, Anamaya Publ., New Delhi, India (2010), pp. 121–129.Google Scholar
  7. 7.
    J. Connor, “The statistical and and strong p-Cesaro convergence of sequences,” Analysis, 8, 207–212 (1998).MathSciNetGoogle Scholar
  8. 8.
    P. Das, “Certain types of open covers and selection principles using ideals,” Houston J. Math., 39, No. 2, 637–650 (2013).MathSciNetMATHGoogle Scholar
  9. 9.
    P. Das, “Some further results on ideal convergence in topological spaces,” Topol. Appl., 159, 2621–2625 (2012).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    P. Das and D. Chandra, “Some further results on ℐ−γ and ℐ−γk-covers,” Topol. Appl., 16, 2401–2410 (2013).Google Scholar
  11. 11.
    P. Das and S. K. Ghosal, “On ℐ-Cauchy nets and completeness,” Topol. Appl., 157, 1152–1156 (2010).Google Scholar
  12. 12.
    P. Das and S. K. Ghosal, “When ℐ-Cauchy nets in complete uniform spaces are I-convergent,” Topol. Appl., 158, 1529–1533 (2011).Google Scholar
  13. 13.
    P. Das and E. Savas, “Some further results on ideal summability of nets in ()-groups,” Positivity, 19, No. 1, 53–63 (2015).Google Scholar
  14. 14.
    O. Dovgoshey, “Tangent spaces to metric spaces and to their subspaces,” Ukr. Mat. Visn., 5, 468–485 (2008).Google Scholar
  15. 15.
    O. Dovgoshey, F. G. Abdullayev, and M. Kücükaslan, “Compactness and boundedness of tangent spaces to metric spaces,” Beitr. Algebra Geom., 51, 100–113 (2010).Google Scholar
  16. 16.
    O. Dovgoshey and O. Martio, “Tangent spaces to metric spaces,” Rep. Math. Helsinki Univ., 480 (2008).Google Scholar
  17. 17.
    I. Farah, “Analytic quotients. Theory of lifting for quotients over analytic ideals on integers,” Mem. Amer. Math. Soc., 148 (2000).Google Scholar
  18. 18.
    H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 41–44 (1951).MathSciNetMATHGoogle Scholar
  19. 19.
    A. R. Freedman and J. J. Sember, “On summing sequences of 0’s and 1’s,” Rocky Mountain J. Math., 11, 419–425 (1981).Google Scholar
  20. 20.
    J. A. Fridy, “On statistical convergence,” Analysis, 5, 301–313 (1985).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    J. Heinonen, Lectures on Analysis on Metric Spaces, Springer (2001).Google Scholar
  22. 22.
    P. Kostyrko, T. Šalát, and W. Wilczyński, “ℐ-convergence,” Real Anal. Exchange, 26, 669–685 (2000/2001).Google Scholar
  23. 23.
    M. Kücuükaslan, U. Deger, and O. Dovgoshey, “On the statistical convergence of metric-valued sequences,” Ukr. Math. Zh., 66, No. 5, 796–805 (2014); English translation: Ukr. Math. J., 66, No. 5, 712–720 (2014).Google Scholar
  24. 24.
    B. K. Lahiri and P. Das, “ℐ and ℐ*-convergence in topological spaces,” Math. Bohem., 130, 153–160 (2005).MathSciNetGoogle Scholar
  25. 25.
    M. Mačaj and T. Šalát, “Statistical convergence of subsequences of a given sequence,” Math. Bohem., 126, 191–208 (2001).MathSciNetMATHGoogle Scholar
  26. 26.
    H. I. Miller, “A measure theoretic subsequence characterization of statistical convergence,” Trans. Amer. Math. Soc., 347, 1811–1819 (1995).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A. Papadopoulos, “Metric spaces, convexity and nonpositive curvature,” Eur. Math. Soc. (2005).Google Scholar
  28. 28.
    T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30, 139–150 (1980).MathSciNetMATHGoogle Scholar
  29. 29.
    H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math., 2, 73–74 (1951).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • P. Das
    • 1
  • E. Savas
    • 2
  1. 1.Jadavpur UniversityWest BengalIndia
  2. 2.Istanbul Commerce UniversityIstanbulTurkey

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